How many small cubes will have red paint on just two faces?

Question

Given a 4cm by 4cm by 4cm cubic block of wood. Suppose all six faces of the cube are painted red and the cube is then cut into 64 small cubes. How many these small cubes will have red paint on just two faces?

Answer 1

You want to find ones with paint on two faces. So you won't want the corners (3 sides painted), you won't want the middle of the faces (1 side painted), and you won't want the middle of the cube (no sides painted). We want the middle cubes of the edges.

 

If we were to divide into 64 other cubes, we would have 4^3=64. In other words we would have 4 cubes by 4 cubes by 4 cubes. Now we take only the middle cubes of the edge. So we take the edges (4 cubes long) and we take out the corners. Now we have 2 cubes on each side that have 2 sides painted.

 

2 cubes on each side times 12 sides is 24 cubes.